Voting has begun in the French presidential election, a ballot which could transform the political colours of France and spark a change in Europe's approach to its debt crisis and austerity measures.

All polls for months have shown the Socialist party candidate, François Hollande, beating the rightwing Nicolas Sarkozy, the most unpopular French president ever to run for re-election. The left is hoping for its first Socialist president since François Mitterrand's re-election in 1988, which would be a rare event in a Europe that has swung to the right in recent years and weakened the left.

If Sarkozy is defeated after one term in office, he would become the 11th European leader to be swept from power since the economic crisis. Hollande has promised to renegotiate Europe's fiscal pact on austerity and shift the focus to growth measures.

Sarkozy has remained defiant, claiming that the polls were mistaken, predicting a higher-than-expected turnout that would play in his favour. In the final days of campaigning, he suggested the poll could be so tight that there may need to be a recount, comparing it to George W Bush's win in 2000 after a Florida recount.

A Sarkozy victory would be a major political turnaround. Despite a narrowing in Hollande's lead in the final days of a vitriolic and brutal campaign, Sarkozy's aides have for days privately admitted that it would take a miracle to turn the odds in his favour.

"He's like a runner – he won't consider it's over until the very end, but I'd say he has one chance in six," one of his inner circle told Reuters before campaigning ended on Friday.

Hollande, spending the weekend in his rural heartland of Corrèze where he is an MP, said he was "confident but not sure". He added: "If I was absolutely sure of the result, I wouldn't have campaigned until the very end."

By 5pm CEST turnout was at 71.96 % – more than in the 22 April first round, but less than in 2007. Polling stations in major cities will close at 8pm CEST with polling estimates based on partial early counts announced by French television at the same time.

Much will depend on turnout and the decision of the 6.4 million voters who voted for the far-right Front National's Marine Le Pen in the first round. Sarkozy needs the vast majority of Le Pen's supporters to vote for him if he is to have a chance at winning. Le Pen said she would vote blank and polls have shown a maximum of 60% of her voters turning to Sarkozy. The president would also need to win over a large number of supporters of the centrist François Bayrou, who took 9% in the first round. Bayrou, in a break from centrist tradition, announced last week he would vote Hollande because he was disgusted at the way in which Sarkozy had veered to the hard-right in a campaign that has seen him stoke anti-immigration and anti-foreigner feeling to court Le Pen's vote.

Whoever becomes the next French president will have no state of grace, leading a country crippled by public debt and in economic crisis, with unemployment nudging a record 10%, a gaping trade-deficit, stuttering growth and declining industry. France's public debt is so high that interest repayments alone account for the second highest state expenditure after education. The rating agency Standard & Poor's this year downgraded France's triple-A credit rating, citing in part its over-high state spending for straining public finances. Both Hollande, a moderate from the centre ground of the Socialist party, and Sarkozy have promised to balance the books – France hasn't had a balanced budget for more than 30 years.

Hollande's manifesto is based on scrapping Sarkozy's tax-breaks for the rich and putting up taxes for high earners to finance what he deems essential spending, including the creation of 60,000 posts in France's under-performing school system. He has pledged to keep the public deficit capped but for his delicate balancing-act to work, he needs a swift return to growth in France, despite economists warning of over-optimistic official growth forecasts that need to be trimmed.

Hollande beat Sarkozy by about half a million votes in the first round of voting on 22 April.

The first round turnout of around 80% was higher than expected and is being closely watched again, with polls suggesting Sarkozy's best chance of an upset comes from an even greater voter turnout on Sunday.